On common fixed and periodic points of commuting functions

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 2 (1998) 269-276

269

ON COMMON

FIXED

AND PERIODIC POINTS OF

COMMUTING FUNCTIONS

ALIASGHAR ALIKHANI-KOOPAEI

MathematicsDepartment University ofIsfahan

Isfahan, Iran

(Received March 8, 1996 and in revised form March 30, 1996)

ABSTRACT. Itisknownthat two commuting continuous functionsonanintervalneednot have acommonfixed point. It isnot known if such two functions have acommon periodicpoint. In this paperwefirst givesomeresults in this direction. Wethen defineanewcontractivecondition, underwhich two continuous functionsmust haveauniquecommonfixed point.

KEY

WORDS AND

PHRASES:

FIXED POINTS,

PERIODIC

POINTS, COMMUTING

FUNC-TIONS

MINIMAL SETS.

1992AMSSUBJECT CLASSIFICATION CODES: 26A16, 47H10.

1.

INTRODUCTION.

For some time there was a rather well known conjecture that if

f

and g are continuous

commuting selfmaps of

[0,1]

(i.e.f(g(x))

g(f(x))),

thentheyhaveacommonfixed point. W.M. Boyce

[1]

and J.P.Huneke

[2]

answeredthis conjecture in thenegative by constructing apair of

commutingcontinuous functionswhichhavenocommonfixed point. It is easy tosee thattheir

pair of commutingcontinuousfunctions haveacommon periodic point.

In

fact for the

Boyce’s

examplezeroisacommonperiodicpoint. Thusonemayconjecture thefollowing:

CONJECTURE 1.1.

I/f

and g are commuting continuous self,naps

of

[0, 1],

then they musthave acommonperiodicpoint.

Even thoughwebelievethat theanswertothe above conjecture is alsonegative,at present wearenot ableto construct acounterexample.

commute,thenthere alwaysexistsapointxsuch thatx

f(z)

g’(x)forsomepositiveinteger n, under the additionalassumptionthat

f

hasacontinuousderivative.

In

thispaperwefirstgive

someresultsin this direction. Thenwedefineaproperty

P1

andshowthatunderthiscondition,

twocommutingcontinuous functions musthaveaunique common fixedpoint.

Webeginwithsomepreliminaries. Theorbitofz underg i.e. the set

{gk(x)

k

_>

0})

and

itsclosurearedenotedbyO(g,z)and

O(g,z),

respectively. ThesetofclusterpointsofO(g,z)is

denoted byw(g,z). A subsetYofIiscalledinvariantunderg ifg(Y) _C Y. A closed,invariant, nonempty subset of

I

iscalledminimal if it containsnopropersubsetthat isalsoclosed,invariant andnonempty.

A

point z iscalledarecurrentpoint ofg if zbelongs tow(g,

z).

Throughout denotes the nfold composition of

f

with itself. Thesets

P(f),

R(f)

and

F(f)

are the sets of periodic points off, recurrent points of

f,

and thefixed points off,respectively.

We

statesome

known facts concerningminimal sets.

Every

closed, invariant, nonempty subsetof

I

containsa minimalset. IfYisaminimalset,then YC_

R(f).

IfYisaminimalset, whichisnotthe orbit ofaperiodic point, then

Y

isperfect.

A

minimalset isnowheredense.

2.

THE MAIN

RESULTS.

THEOREM 2.1. Let

f

and g be two commuting continuousselfmaps

of

the unit interval.

If f

andg have no commonperiodicpoints, then

for

any twopositive integers rn andn the set

A,,

{" f"()=

()}

==o=ta.

PROOF. First weshow that

A,,,n

is not empty. If forsome rn and n,

A,,n

is empty,the continuity of

f

and g permitsustoassumewithoutlossof generality,that

ff’(z)

<

g’(x)

(2.1)

for all xEI. Sincegin(l)

<

1, theset

S

{x

E

I’g"(x) <

x}

is notempty. Thus,since

S

is closed, ithas aminimumelementc. Clearly,c

g’(c).

Hence

f’(c)

f"(g"(c))

g"(f’(c)),

so that

f"(c)

q. _{S. Consequently}

f’(c)

>

c

g’(c).

Since

f"(c) >_ g,"(c)

contradicts

(2.1),

the assertion that

A,,,,,

is emptyis false. Now suppose that z A

....

Then

f’(z)

g"(z) hence,

f"(f(x))

f(g’(x))

g(f(x))

and

f"(g(x))

g(f(x))

g(g’(x))

g"(g(x)).

Thus

ifx

A

....

{f(x),g(x)}

C_

A,,n

and

fP"(z)

gW(x)

for each positiveintegerp. Fromthis it

follows that

O(g,x)

is contained in

A,,,

whenever x G

A,,.

The set

O(g,x)

is an invariant set underg.

Suppose

A1

is aminimalsetcontained in

O(g,x).

Since

f

andghaveno common

periodic points, the set

A1

cannot be theorbit ofa periodic point. Hence

A

is a perfect set

contained in

A,,,,

implyingthat

A,,,,

isuncountable.

COROLLARY2.1. Let

f

and g be two commutingcontinuousselfmaps

of

theunit interval.

PERIODIC POINTS OF COMMUTING FUNCTIONS 271

Fixed points of contractivetypemappings have been studied byanumber ofauthors. B.E. Rhoades

[4]

has investigatedacomparisonof different kindsofcontractivedefinitionsin the liter-ature. Thefollowingtheoremis in this direction.

DEFINITION 2.1. Let

X

bea compactmetricspace. The

function

h"

X

X

[0,

issaid tohave property

P

if

it

satisfies

thefollowingconditions:

(i)"

h(x,

y)

0

if

and only

if

x y,

(ii)"

if

limn__xn z0,lirn_ y Yo, and lirn_

h(x,,, y)

O, thenxo yo.

THEOREM 2.2. Let

f

and9 be selfmaps

of

theunit intervalandlet h I

I

be a

function

havingproperty

P1.

Suppose

g is continuous on

I

and

A

is a nonempty closed

g-invariantsubset

of

F(f).

If

there ezists areal number a, 0

<

a

<

such that

for

allx andy in

F(f),

f

andgsatisfy the following inequality:

h(fz, f) < a.max{h(gz,gy),h(gz, fz),h(gy, fy),

h(gy,

f

x), h(f

x,gy)

},

(2.2)

then

f

andg have aunique common

fixed

point.

PROOF. Firstweshow thatany twofunctions

f

andgsatisfying inequality

(2.2),

haveat

mostonecommonfixed point.

To

seethis,onthe contrary, suppose ql andqz are two different commonfixedpoints of

f

andg. Then using

(2.2)

weget

h(ql,q)

h(f

ql,

f

q)

_<

a-

max{

h(gq gq

),

h(gqa

f

q

),

h(gq,

f

qz

),

h(gq,

f

q

),

h(

f

q gq implying

h(q,q.)

<_

a.

max{h(q,q),h(q,q),h(q,q),h(q,q)},

a.

max{h(q,,q2), h(q,ql)}.

In

asimilarmanner wemayshow

h(q2,

ql)

<

a.max{h(q,q_), h(q,q)},

henceweshall have

max{h(q,q2), h(q:,q)}

<

a.max{h(qi,q2), h(q2,

q)}.

This isimpossible unless

max{

h(q, q.), h(q2,

ql)}

O,whichimpliesthath(q, q=)

O,

and hence

q q2, acontradiction. Nowweshow that

f

and ghavea commonfixed point. Without loss

nonempty subset of

A,

and that

g2(B)

B. If for some yo 6_.

B,

f(yo)

g(yo),

then yo is a commonfixed point of

f

and g,sowemay assume that f(y) g(y)for allyE B. Let y bean arbitrary point of

B,

and x2 E

B

besuchthat

g(x)

yl Since

g(y)

#

f(y)for all y

B,

ifwe letx yl andy2

g(x),

theng(y2) yl

f(y).

Sinceh(gy, fy2)

h(fyl, fy),

from

inequality

(2.2)

wehave

h(fyl,fy)

<_

a.

max{.h(gyl,gy2),h(gyl,fyl),h(gy,fy),

h(gy_,

f

yl

), h(f

y,,

gy)

}

a.

max{h(gy,gy),

h(gy, fy,), h(gy, fy),

0))

a.

m{

h(gy,

fl),

h(gy2,

fy2)

),

implyingh(gy:,

fy:)

a.h(gy,

fyl).

Ifwedefine

"

B

[0,

)

by

(x)

h(gx,

fx),

thenwe

have

(y)

a-(yx). Suppose

x3

B

be such that

g2(x3)

y. Bytakingx y2andyz

g(x3),

similarlywe canshowthat

(yz)

a.

(y2)

a2"

(y). By

_{repeatingthisprocess weobtn a}

boundedsequence

{y}

in B.

Suppose {y,}=

isa convergent subsequence of

{y,}.

d

lim_y y0 Since gis continuouson

I,

lira

h(gy,y)

lira

h(gy,fy)

k

lira

(y,)

<

lira a

"-

(y)

0.

So

g(yo)

yo,andy0

B

F(f),

hencey0isacommonfixed pointof

f

and g.

REMARK

2.1.There isnothingin theprf of Threm 2.2that requiresthe underlying

space to betheunit intervM. Howeverinourproofthecompactnessof the unit intervM isused. Thereforeone canlet the underlyingspace to beacompactmetric space. Alsocondition

P

isnot anecessary condition anditmay bereplacedwithweberconditionswhen

f

and g enice. For exampleifh,

f

areniceenoughthat thefunction defined inTheorem 2.2attains itsminimum

at somepoint y

B,

then wecan findapointy

B

suchthat

(y:) < (y)

acontradiction, so

f

dg must haveacommonfixed point. Ifwearenotconcernedabout theuniquenessof the commonfixed point, similartotheproofof Threm 2.2wehave thefollowing.

THEOREM 2.3.

Suppose

f

andg am twoselfmaps

of

acompactmetricspace

X

andlet

h

X

x

X

[0, )

bea

function

havingproperty

P.

Suppose

alsothat g is continuous on

X

and

A

is a nonempty closedg-invariant subset

of

F(f).

If

there exists a real number a, 0 a

<

such that

for

all x andyin

A,

f

and g satisfythefollowing inequality:

h(fx,

fy) a.

max{h(gx,

gy),h(gx, fx),h(gy, fy),

PERIODIC POINTS OF COMMUING FUNCTIONS 273

then

f

andg haveacommon

fized

point.

The conclusion of Theorem 2.3isnotvalidifinequality

(2.3)

isreplacedwith

h(yx, fy)

< max{h(gz, gy),h(gx, fx),h(gy,

fy), h(gy,

fx),h(fx, gy)}.

Thefollowing exampleillustratesthis.

EXAMPLE

2.1. Let

f

and9bedefinedontheunitintervalasfollows"

(8x

+

15)/24

0

<

x

<

3/4,

(-32x

+

31)/8

3/4

<

z

<_ 13/16,

2x-

13/16

<

x

_<

7/8,

3/4

7/8

<

x

<_

1,

f(x)

(8x

+

39)/48

0

<_

x

< 3/4,

3/4

x

3/4,

(-32z

+

39)/16

3/4 <

x

<

13/16,

x

13/16

<

x

< 7/8,

7/8

7/8

<

z

<

1.

Define

h(x, y)" I

x

I

-,

[0, oo)

as

h(z,y)=

0 if z y,

z+y

ifx#y.

(l+g(x))/2

if

x#3/4,

Onecaneasily check that h haspropertyP1,

f(x)

3/4

if x

3/4,

and

F(f)

{3/4} [J[13/16, 7/8].

Let

A

{3/4, 7/8},

then

A

is aclosed g-invariant subset of

F(f)

and foreveryx

-

y inAwehave

h(fx,

fy)

max{h(gx,

gy),h(gx,

fz),h(gy,

fy),

h(gy,

fx),h(fx,

gy)} <

13/.

Itis easy toseethat

f

andg donot haveacommonfixedpoint.

THEOREM 2.4.

Suppose

f

andg are twoselfmaps

of

a compactmetricspace

X

with g

continuous, and let h

X

X

[0, oo)

is a

function

havingproperty

P1.

If for

allx

#

y in

X,

f

andg satisfythefollowing inequality:

h(fx,

fy)

<

max{h(gx, gy), h(gx,

fx),

h(gy, fy),

h(gy,

fx),h(fz,

gy)),

(2.4)

then one

of

thefollowing holds:

(i

O

oreverynonempty closed

9-invariant

subset

of

F(f)

contains aperfectminimalset

B

such that the

functions 1(z)

h(gz,

z)

and

O(z)

h(z,gz),

do not attain their minimum or mazimum onB.

PROOF. Frominequality

(2.4)

it followsthat foreach zE

F(f)

with z

9(z),

wehave h(gz,

z)

<

h(9z,

gz). Thisimpliesthat

F(f)

cannotcontain anyperiodicpoint of9 withperiod greaterthan1. Let

A

beanonempty closed

9-invariant

subset of

F(f),

thenitshouldcontaina minimal setB. Ifthisminimal set isfinite,

f

and9havea commonfixedpoint. Otherwise,

B

is aperfect set with

g(B)

B. Sincefor ech x

.

B,

h(gx,

x) < h(gx,

gx),

thefunctions and

cannot attain their maximumorminimumonB.

:LEMMA 2.1.

Suppose

h-J0,1]

[0,1]

[0, or)

is alowersemicontinuous

function

such that

h(x,

y)

0

if

andonly

if

z y. Thenh

satisfies

property

PROOF.

Suppose

lim,,...oo

xn

z0, lim,, y, y0, and lim,...oo

h(x,,,y,,)

0. Weneed

toshowthatx0 yo. Onthecontrarysupposezo y0. Let dist

{

(z0,

yo),

{ (z, z).

z

[0,

]}

and

B1

B(z0,

)=

B[z0,

],

wherez0

(x0,y0).

Let ebeanarbitrary positive number. Then

there existsapositiveintegerA/"such that for alln

>

A/’,(z,,,

y,,) fi

Bx,

and

h(x,,

y,,)

l<

e. Since his lowersemicontinuousand

B1

is compact,itattains its minimum at somepoint

(s,t)

in

B1.

Hence

h(s,t) [=[minimum

of

h(x,y)

on

Bx

I<l

h(xf,y.q)

1<

e,implying

h(,t)

0. Thuss t, acontradiction.

THEOREM 2.5. Let

f

andg be commutingselfmaps

of[O,

1]

with g continuous.

If

A

is a nonemptyclosed g-invariantsubset

of

F(f),

thenone

of

the following holds:

(i)

either

P(f)

P(g)

,

(ii}

orthereisaperfectset

Aa

C_

(AflR(g)flP(f)).

PROOF. TakeXo A. Foreverypositive integern,g"(xo) A. Thus

O(g,

xo)

C

A

im-plyingw(g,

xo)

C_ A. Clearlyw(g,

zo)

isnonemptyand is invariant underg. Thus itcontains aminimal g-invariant subset

A.

Obviously

A

C_

(Af’lR(g)).

If

A1

isafiniteset,thenit isthe

orbitofaperiodicpointofg,implying

1

(P(g)fqA)

C_

p(.f)fp(g).

Otherwise

A

is aperfect set containedin

A

f’l

R(g).

REMARK

2.2.Asinthe proof of Theorem2.5,if

f

and ghavenocommonperiodicpoint

then thereexitsaperfectset

A1

contained in

A

Iq

R(g)

f’l P(f).

Since

f

and g donothavecommon

periodic pointsand,foracontinuous functiononacompactinterval

P(f)

P(g)

seeCovenand Hedlund

[5]

),

wehave

A

C_

A f’lR(g)

Af’IP(g)= A[P(g)

\

P(g)].

Fromthis itfollowsthat

ifP(g)

\

P(g)doesnot contain aperfect set, then

f

andg musthaveacommonperiodic point.

In

particular, fortwocommutingcontinuousfunctions ifeither of

P(g)

\

P(g)

or

P(y)

\ P(f)

do notcontainaperfect set, then

f

andg must haveacommonperiodic point.

PERIODIC POINTS OF COMMUTING FUNCTIONS 275

1,2,4,8,. ,7-8,5-8,3-8,...,7-4,5-4,3-4,...,7- 2,5- 2,3- 2,...,7,5,3.

A.N. Sarkowskii has proven that if rn is to the left of n in the above ordering and

f

has a periodic point ofperiod n, then

f

must have a periodic point of period m

(see

Stefan

[6]).

Let

P,(f)

{x

E

[0,1]

f"(x)

x}.

The Sarkovskii’s theorem immediately implies that if

Pl(f)

P2(f),

then

Pl(f)

P,,(f)

for every n

>

1.

Suppose

f

and g are two commuting

continuousselfmapsoftheunitinterval whichdonot havea commonperiodicpoint. We claim that

f

andgshouldhave periodic points ofallevenorders. Toseethis,onthecontrarysuppose

that n 2k-

r(r

_odd,k

_>

1)

_and

f

_{hasno periodic pointofordern.}

By

_{Sarkowskii’stheorem}

P2.r(f)

Pr(f)

implying

Pl(f)

P2(f),

hence

Pn(ff

Pl(ff

for anyn

>

1. Thuswehave

P(f)

(J’=l

P,,(ff)

P,(f)

whichimplies

P(f)

\ P(ff)

}. Since g commuteswith

f,

italso commuteswith

ft.

Thusgand

ff

shouldhaveacommonperiodic pointwhich isalsoa common periodic point of

f

andg. Wemayinterchange the roles of

f

andg. Thisimplies that either

f

and

g haveacommonperiodic pointorbothhavearich orbitstructure. Byarich orbitstructurewe meanthat they haveahomoclinic point and positive topological entropy see Block

[7] ).

Thus wehave thefollowing:

THEOREM 2.6.

Suppose

f

andg are two commuting continuous selfmaps

of

the unit interval. Thenone

of

the following holds:

(i)

either

P(f)f’l

P(g)

,

(ii)

orboth

f

andg have periodicpoints

of

allevenorders.

ACKNOWLEDGMENT: wouldlike to express my sincerethankstoprofessorB.E.Rhoades for his valuable suggestions inimproving the paper, especially on the comments that improved Theorems2.2and 2.3. Thiswork alsowascompletedwhilethe authorwasspendinghis sabbatical

at the University of North Carolinaat Wilmington. The hospitality ofits

Department

of

Mathe-maticsas wellasthefinancialsupport of Isfahan Universityaregreatly acknowledged.

AUTHOR’S PRESENT ADDRESS: Mathematical Sciences

Department,

The University of

North Carolina atWilmington, Wilmington,NC28403-3297.

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W.M.,

Commutingfunctionswithnocommonfixed point,Trans. Amer. Math. Soc.

137(1969),

77-92.

HUNEKE,

J.P.,

Oncommonfixed pointsofcommuting functionson aninterval,Trans. Amer. Math. Soc.

139(1969),

371-381.

SCHWARTZ,

A.J.,

Commonperiodic points of commuting functions, Michigan Math. J.

4.

RHOADES, B.E., A

comparisonof various definitionsofcontractivemappings,Trans. Amer. Math. Soc.

226(1977),

257-290.

5.

COVEN,

E.M.,

and

HEDLUND,

G.

A., P

R

for maps of the interval,Proc. Amer. Math.

Soc.

79(1980),

316-318.

6.

STEFAN, P., A

theoremofSarkowskiiontheexistenceof periodicorbits of continuous

endomorphismsoftherealline,Comm. Math. Phys.

54(1977),

237-248.

7.

BLOCK, L.,

Homoclinic points ofmappings of the interval,Proc. Amer. Math. Soc. 72